I am not too familiar with the Pi operating system and what it can and can't do. I am running the ADSB along with a few other feeders with the ADSB as the initial install. Can I install other programs on the Pi, like my weather information and associated programs or will that cause a conflict with the feeders?
The Pi operating system is based on Debian Linux, hence the name Raspbian. Installing other programs shouldn't conflict with feeding ADS-B but they might max out your Pi's processor and/or RAM depending on what model of Pi you have. An 8GB Pi 4 should be able to handle anything that a similarly-specded PC can, with the limitation being your SD card's capacity. A Pi Zero with 512KB of RAM, no.
There are no similarly specced PCs as none use and sd-card as storage. Anyhow you can run lots of stuff. Lots of stuff made for the pi will take over the webserver (lighttpd) completely and break the local interface. You'd have to know the weather info software in question .... to know if it interferes.
I meant regarding processor speed, number of cores and amount of RAM, not storage. I know of people using Pi 4s as desktop PC replacements and they stream Netflix videos on them, for example.